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Laws of the iterated logarithm for the range of random walks in two and three dimensions. (English) Zbl 1031.60031

Let \((S_n)\) be a random walk in \(Z^d\), i.e. the sum of i.i.d. centered random variables \(X_i\) taking values in \(Z^d\). Let \(R_n\) be the range of the random walk, i.e. \(R_n\) is the number of distinct sites visited by \(S_0, S_1, \dots ,S_n\). For \(d=3\), under an additional assumption on the moments of \(X_1\), the authors prove an almost sure invariance principle for \(R_n\). As a corollary, they obtain laws of the iterated logarithm. For \(d=2\), assuming that the two coordinates of \(X_1\) are independent and bounded, they prove a law of the iterated logarithm for the range \(R_n\).

MSC:

60G50 Sums of independent random variables; random walks
60F15 Strong limit theorems
60G17 Sample path properties
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
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[24] STORRS, CONNECTICUT 06269 E-MAIL: bass@math.uconn.edu RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES Ky OTO UNIVERSITY Ky OTO 606-8502 JAPAN E-MAIL: kumagai@kurims.ky oto-u.ac.jp
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